It has been known for sometime that the Abelian Sandpile is an
excellent model of Self Organized Criticality. It has been extensively
studied from many points of view. In particular, from the algebraic point
of view, the recurrent elements have the structure of a finite Abelian
group, whose structure has been studied in the literature. More generally
the set of all stable configurations has the structure of a finite Abelian
semigroup. The purpose of this talk is to give a detailed introduction to
the structure of this semigroup. We prove that the semigroup is a
nilpotent
ideal extension of the group of recurrent elements. This is the algebraic
equivalent of the fact that every long enough sequence of avalanches leads
to a recurrent state. We compute the structure of the nilpotent part of
the
semigroup in the case of a one dimensional sandpile. This gives detailed
information about the transient elements of the model including the length
of the longest non-recurrent sequence.
Sponsored in part by the FCT approved projects POCTI 32817/99 and POCTI/MAT/37670/2001 in participation with the European Community Fund FEDER and by FCT through Centro de Matemática da Universidade do Porto. Also sponsored in part by FCT, the Faculdade de Ciências da Universidade do Porto, Programa Operacional Ciência, Tecnologia, Inovação do Quadro Comunitário de Apoio III, and by Caixa Geral de Depósitos.